Parametrization of the attainable set for a nonlinear control model of a biochemical process

doi:10.3934/mbe.2013.10.1067

Mathematical Biosciences and Engineering, 2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067. Primary: 49J15, 49N90; Secondary: 93C10, 93C95.

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Parametrization of the attainable set for a nonlinear control model of a biochemical process

1. Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204
2. Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992
3. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona

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Abstract

In this paper, we study a three-dimensional nonlinear model of a controllable reaction$ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecifiednonlinear function. A model of this type describes a variety of real-life processes in chemicalkinetics and biology; in this paper our particular interests is in its application to wastewater biotreatment. For this control model, we analytically study the corresponding attainableset and parameterize it by the moments of switching of piecewise constant control functions.This allows us to visualize the attainable sets using a numerical procedure.
These analytical results generalize the earlier findings, which were obtained for a trilinearreaction rate (which corresponds to the law of mass action) and reportedin [18,19], to the case of a general rate of reaction. These results allow toreduce the problem of constructing the optimal control to a straightforward constrained finitedimensional optimization problem.

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Keywords: waste water biotreatment; Environmental problem; nonlinear reaction rate; attainable set.

Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences and Engineering, 2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067

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